11 - Calculus

Question

Suppose you wanted to calculate the upper sum approximation for the area between the graph of f(x) = (x − 1)2 and the x-axis from x = 0 to x = 2. List all of the values Mk used for (a) n = 2 rectangles, (b) n = 3 rectangles, and (c) n = 4 rectangles. Sketch graphs of your rectangles to illustrate your answers 

Step-by-Step Solution

Answer

As below Steps we got the solution 

Part(a) Step1: Given Information

The function is,

$f\left(x\right)=(x-1{)}^2$

The objective is to list the values of $M_k$ used for $n=2$ rectangles while calculating the upper sum approximation for the area of the graph and the x-axis from $x=0 to x=2$.

Step2: Calculation

The upper sum defined for n rectangles on $[a, b]$ is $\sum _{k=1}^{n}f\left(M_k\right)\Delta x$

Where, $M_k$ is the maximum value of height chosen in the interval $[x_{k-1},$$x_k]$

$\Delta x=\frac{b-a}{n},x_k=a+k\Delta x$

Consider the following figure,

The maximum values $M_k$ occurs at x=0 and $\mathrm{x}=2$.

Therefore, the values are $M_1=0$, $M_2=2$

Part(b) Step1: Given Information

The function is,

$f\left(x\right)=(x-1{)}^2$

The objective is to list the values of $M_k$ used for n=3 rectangles while calculating the upper sum approximation for the area of the graph and the x-axis from x=0 to x=2.

Step2: Calculation

The upper sum defined for n rectangles on$[a,b]$ is $\sum _{k=1}^{n}f\left(M_k\right)\Delta x$

Where, $M_k$ is the maximum value of height chosen in the interval $\left[x_{k-1},x_k\right]$,

$\Delta x=\frac{b-a}{n},x_k=a+k\Delta x$

Consider the following figure,

The maximum values Are at x=0, x=2/3  or $\frac{4}{3}$ and x=2.

Therefore, the values are $M_1=0,M_2=\frac{2}{3}$ or $\frac{4}{3}$, $M_3=2$.

Part(c) Step1: Given information

The function is,

$f\left(x\right)=(x-1{)}^2$

The objective is to list the values of $M_k$ used for $\mathrm{n}=4$ rectangles while calculating the upper sum approximation for the area of the graph and the x-axis from $x=0 to x=2.$.

Step2: Calculation

The upper sum defined for $\mathrm{n}$ rectangles on $[\mathrm{a},\mathrm{b}]$ is $\sum _{k=1}^{n}f\left(M_k\right)\Delta x$

Where,$M_k$ is the maximum value of height chosen in the interval $\left[x_{k-1},x_k\right]$,

$\Delta x=\frac{b-a}{n},x_k=a+k\Delta x.$

Consider the following figure,

The maximum values $M_k$ occurs at $\mathrm{x}=0,x=\frac{1}{2},x=\frac{3}{2}$and $\mathrm{x}=2$.

Therefore, the values are $M_1=0,M_2=\frac{1}{2},M_3=\frac{3}{2},$$M_{34}=2$